The second Stekloff eigenvalue and energy dissipation
           inequalities for functionals with surface energy.

                                   by

                           Robert Lipton, WPI
                            and Univ Arizona

                JWB 208, 3:20pm  Friday, April 25, 1997

                                Abstract


 A functional with both bulk and interfacial surface energy is
considered. It corresponds to the energy dissipated inside a multi-phase
electrical conductor in the presence of an electrical contact resistance
at phase interfaces. The effect of embedding a conducting particle
inside a region of lesser conductivity is investigated. We find the
criterion that determines when the increase in surface energy matches or
exceeds the reduction in bulk energy associated with the particle. This
criterion is general and applies to any particle with Lipschitz
continuous boundary. It is given in terms of the second Stekloff
eigenvalue of the particle. The inequalities of Bramble and Payne are
applied to estimate the second Stekloff eigenvalue for starlike domains.
The estimates are used together with the energy dissipation inequalities
to prove existence of energy minimizing configurations of particles and
to provide rigorous rules of thumb for the design of particle reinforced
composites.


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